Complex projective space

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In mathematics, complex projective space, P(Cⁿ⁺¹), P_n(C) or CPⁿ, is the projective space of (complex) lines in Cⁿ⁺¹. The case n = 1 gives the Riemann sphere (also called the complex projective line), and the case n = 2 the complex projective plane.

1 Properties
2 Construction
3 Topology
4 Geometry
5 See also

Properties

CPⁿ is a complex manifold of complex dimension n, so it has real dimension 2n. It is a special case of a Grassmannian, and is a homogeneous space for various Lie groups. It is a Kähler manifold carrying the Fubini-Study metric, which is essentially determined by symmetry properties. It also plays a central role in algebraic geometry; by Chow's theorem, any compact complex submanifold of CPⁿ is the zero locus of a finite number of polynomials, and is thus a projective algebraic variety.

Construction

Complex projective space is a complex manifold that may be described by n+1 complex coordinates as

$(z_1,z_2,\ldots,z_{n+1}) \in \mathbb{C}^{n+1}, \qquad (z_1,z_2,\ldots,z_{n+1})\neq (0,0,\ldots,0)$

where the tuples differing by an overall rescaling are identified:

$(z_1,z_2,\ldots,z_{n+1}) \equiv (\lambda z_1,\lambda z_2, \ldots,\lambda z_{n+1}); \quad \lambda\in \mathbb{C},\qquad \lambda \neq 0.$

That is, these are homogeneous coordinates in the traditional sense of projective geometry.

One may also regard CPⁿ as a quotient of the unit 2n+1 sphere in Cⁿ⁺¹ under the action of U(1):

CPⁿ = S²ⁿ⁺¹/U(1)

This is because every line in Cⁿ⁺¹ intersects the unit sphere in a circle. By first projecting to the unit sphere and then identifying under the natural action of U(1) one obtains CPⁿ. For n=1 this construction yields the classical Hopf bundle.

Topology

Point-set topology

Projective space is compact and connected, being a quotient of a compact, connected space.

Homotopy groups

From the fiber bundle

$S^1 \to S^{2n+1} \to \mathbf{CP}^n$

or more suggestively

$U(1) \to S^{2n+1} \to \mathbf{CP}^n$

$\mathbf{CP}^n$ is simply connected, has $\pi_2(\mathbf{CP}^n) = \mathbf{Z}$ , and higher homotopy agrees with that of $S 2 n + 1$

Homology

In general, the algebraic topology of CPⁿ is based on the rank of the homology groups being zero in odd dimensions; also H_2i(CPⁿ, Z) is infinite cyclic for i = 0 to n. Therefore the Betti numbers run

1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...

The Euler characteristic of CPⁿ is therefore n+1. By Poincaré duality the same is true for the ranks of the cohomology groups. In the case of cohomology, one can go further, and identify the graded ring structure, for cup product; the generator of H²(CPⁿ, Z) is the class associated to a hyperplane, and this is a ring generator, so that the ring is isomorphic with

Z[T]/(Tⁿ⁺¹),

with T a degree two generator. This implies also that the Hodge number h^i,i = 1, and all the others are zero.

Classifying space

There is a space CP^∞ which, in a sense, is the limit of CPⁿ as n → ∞. It is BU(1), the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class. CP^∞ is also the same as the infinite-dimensional projective unitary group; see that article for additional properties and discussion.

Geometry

The natural metric on $\mathbf{CP}^n$ is the Fubini-Study metric, and its isometry group is the projective unitary group $PU(n + 1)$ , where the stabilizer of a point is $\mbox{P}(1\times \mbox{U}(n)) \cong \mbox{PU}(n)$ .

It is a Hermitian symmetric space, $U(n+1)/(U(1) \times U(n)) \cong SU(n+1)/S(U(1) \times U(n))$ .

It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that isn't a sphere (or covered by a sphere): by the 1/4-pinched sphere theorem, any complete, simply connected manifold with curvature strictly between 1/4 and 1 is homeomorphic to the sphere. Complex projective space shows that 1/4 is sharp.